Automorphisms and the fundamental operators associated with the symmetrized tridisc
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The automorphisms of the symmetrized polydisc $\mathbb{G}_n$ are well-known and are given in the coordinates of the polydisc in Edigarian and Zwonek (Arch. Math. 84 (2005) 364–374). We find an explicit formula for the automorphisms of Gn in its own coordinates. If $\tau$ is an automorphism of $\mathbb{G}_n$, then $\tau (S_1,\ldots , S_{n−1}, P)$ is a $\tau_n$-contraction, where a $\tau_n$-contraction is a commuting $n$-tuple of Hilbert space operators for which the closed symmetrized polydisc $\tau_n$ is a spectral set. Corresponding to every $\tau_n$-contraction $(S_1,\ldots , S_{n−1}, P)$, there exist $n − 1$ unique operators $A_1,\ldots , A_{n−1}$ such that $$S_i − S^∗_{n−i} P = D_P A_i D_P,\quad D_P = (I − P^∗P)^{1/2},$$ for $i = 1,\ldots , n − 1$. This unique $(n − 1)$-tuple $(A_1,\ldots , A_{n−1})$, which is called the fundamental operator tuple or $\mathcal{F}_O$-tuple of $(S_1, \ldots , S_{n−1}, P)$ in the literature, plays central role in every section of operator theory on $\tau_n$. We find an explicit form of the $\mathcal{F}_O$ tuple of $\tau (S_1,\ldots , S_{n−1}, P)$ when $n = 3$. We show by an example that a $\tau_n$-contraction may not have commuting $\mathcal{F}_O$-tuple. Also, we obtain a necessary and sufficient condition under which two $\tau_n$-contractions are unitarily equivalent.
BAPPA BISAI^{1} ^{} SOURAV PAL^{1} ^{}
Volume 131, 2021
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